Cone (geometry)

A cone is a three-dimensionalgeometric shape that tapers smoothly from a flat, round base to a point called the apex or vertex. More precisely, it is the solid figure bounded by a plane base and the surface (called the lateral surface) formed by the locus of all straight line segments joining the apex to the perimeter of the base. The term "cone" sometimes refers just to the surface of this solid figure, or just to the lateral surface.

The axis of a cone is the straight line (if any), passing through the apex, about which the lateral surface has a rotational symmetry.

In general, the base may be any shape, and the apex may lie anywhere (though it is often assumed that the base is bounded and has nonzero area, and that the apex lies outside the plane of the base). For example, a pyramid is technically a cone with a polygonal base. In common usage in elementary geometry, however, cones are assumed to be right circular, where right means that the axis passes through the centre of the base (suitably defined) at right angles to its plane, and circular means that the base is a circle. Contrasted with right cones are oblique cones, in which the axis does not pass perpendicularly through the centre of the base.

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In mathematical usage, the word "cone" is used also for an infinite cone, the union of any set of half-lines that start at a common apex point. This kind of cone does not have a bounding base, and extends to infinity. A doubly infinite cone, or double cone, is the union of any set of straight lines that pass through a common apex point, and therefore extends symmetrically on both sides of the apex.

The boundary of an infinite or doubly infinite cone is a conical surface, and the intersection of a plane with this surface is a conic section. For infinite cones, the word axis again usually refers to the axis of rotational symmetry (if any). 1/2 of a double cone is called a nappe.

The perimeter of the base of a cone is called the directrix, and each of the line segments between the directrix and apex is a generatrix of the lateral surface. (For the connection between this sense of the term "directrix" and the directrix of a conic section, see Dandelin spheres.)

The base radius of a circular cone is the radius of its base; often this is simply called the radius of the cone. The aperture of a right circular cone is the maximum angle between two generatrix lines; if the generatrix makes an angle θ to the axis, the aperture is 2θ.

A cone with its apex cut off by a plane parallel to its base is called a truncated cone or frustum. An elliptical cone is a cone with an elliptical base. A generalized cone is the surface created by the set of lines passing through a vertex and every point on a boundary (also see visual hull).

The lateral area of a cone may be derived by approximating it with an n-sided pyramid and taking the limit as n approaches infinity.

Let

where n is the number of sides of the pyramid's base. The pyramid's lateral area is given by

Taking the limit as n approaches infinity, we obtain the lateral area of the cone:

Since R is constant, this can be rewritten as:

Each limit in this expression can be evaluated separately.

1.

This expression can be proved to be equal to . Let L be the length of an edge in the pyramid's polygonal base. The relation between L and the polygon's radius R is

The perimeter of the polygon is given by:

Therefore, the perimeter of a regular polygon is:

Taking the limit as n approaches infinity, we obtain the perimeter of the circular base of the cone:

But since the ratio of a circle's perimeter to its diameter 2R is equal to , we have:

Since 2R is constant and non-zero, this expression can be divided by 2R both in the nominator and the denominator, giving:

2.

Since , we have:

So, the original square root becomes

This expression is equal to the hypotenuse of the right triangle whose height is the height of the cone, and whose base is the radius of the cone. Therefore, is equal to the length of the line from the apex of the cone to its circular edge.